Question

In the following exercises, graph each ellipse.

$\frac{x^2}{9}+\frac{y^2}{25}=1.$

Short answer

The graph of the ellipse is

Step-by-step solution

Step 1. Given information

The equation of the ellipse is$\frac{x^2}{9}+\frac{y^2}{25}=1.$

Step 2. Conceptused

The standard form of the ellipse with

Center$(0,0)$is

localid="1645528234928" $$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ a>b\Rightarrow \text{majoraxisis}x-\text{axiswithendpoints}\left(-a,0\right)\text{,(a,0)} \\ b>a\Rightarrow \text{majoraxisis}y-\text{axis.withendpoints}\left((0,-b\right)\text{,(0,b)} \end{gathered}$$

Step 3. Find the center and major axis

$\frac{x^2}{9}+\frac{y^2}{25}=1$

Compare with the standard form of the ellipse

localid="1645434250033" $$\begin{gathered} \text{center}(0,0) \\ 25>9\text{and} \\ {a}^2=9\Rightarrow a=\pm 3 \\ {b}^2=25\Rightarrow b=\pm 5 \end{gathered}$$

Major axis:

On $y-$axis with endpoints$(0,-5),(0,5).$

Minor axis:

On $x-$axis with endpoints$(-3,0),(3,0).$

Step 4. Sketch the ellipse

The graph of the ellipse is shown below.